General OneDimensional AdvectiveDiffusive Transport
Description
Estimates the transport term (i.e. the rate of change of a concentration due to diffusion and advection) in a onedimensional model of a liquid (volume fraction constant and equal to one) or in a porous medium (volume fraction variable and lower than one).
The interfaces between grid cells can have a variable crosssectional area, e.g. when modelling spherical or cylindrical geometries (see example).
Usage
1 2 3 4 5 
Arguments
C 
concentration, expressed per unit of phase volume, defined at the centre of each grid cell. A vector of length N [M/L3] 
C.up 
concentration at upstream boundary. One value [M/L3] 
C.down 
concentration at downstream boundary. One value [M/L3] 
flux.up 
flux across the upstream boundary, positive = INTO model
domain. One value, expressed per unit of total surface [M/L2/T].
If 
flux.down 
flux across the downstream boundary, positive = OUT
of model domain. One value, expressed per unit of total surface [M/L2/T].
If 
a.bl.up 
convective transfer coefficient across the upstream
boundary layer. 
a.bl.down 
convective transfer coefficient across the downstream
boundary layer (L). 
D 
diffusion coefficient, defined on grid cell interfaces.
One value, a vector of length N+1 [L2/T], or a 
v 
advective velocity, defined on the grid cell
interfaces. Can be positive (downstream flow) or negative (upstream flow).
One value, a vector of length N+1 [L/T], or a 
AFDW 
weight used in the finite difference scheme for advection,
defined on grid cell interfaces; backward = 1, centred = 0.5, forward = 0;
default is backward. One value, a vector of length N+1, or a

VF 
Volume fraction defined at the grid cell interfaces. One value,
a vector of length N+1, or a 
A 
Interface area defined at the grid cell interfaces. One value,
a vector of length N+1, or a 
dx 
distance between adjacent cell interfaces (thickness of grid
cells). One value, a vector of length N, or a 
full.check 
logical flag enabling a full check of the consistency
of the arguments (default = 
full.output 
logical flag enabling a full return of the output
(default = 
Details
The boundary conditions are either
(1) zerogradient.
(2) fixed concentration.
(3) convective boundary layer.
(4) fixed flux.
The above order also shows the priority. The default condition is the zero gradient. The fixed concentration condition overrules the zero gradient. The convective boundary layer condition overrules the fixed concentration and zero gradient. The fixed flux overrules all other specifications.
Ensure that the boundary conditions are well defined: for instance, it does not make sense to specify an influx in a boundary cell with the advection velocity pointing outward.
Transport properties:
The diffusion coefficient (D
),
the advective velocity (v
),
the volume fraction (VF), the interface surface (A
),
and the advective finite difference weight (AFDW
)
can either be specified as one value, a vector or a 1D property list
as generated by setup.prop.1D
.
When a vector, this vector must be of length N+1, defined at all grid cell interfaces, including the upper and lower boundary.
The finite difference grid (dx
) is specified either as
one value, a vector or a 1D grid list, as generated by setup.grid.1D
.
Value
dC 
the rate of change of the concentration C due to transport, defined in the centre of each grid cell. The rate of change is expressed per unit of phase volume [M/L3/T] 
C.up 
concentration at the upstream interface. One value [M/L3]
only when ( 
C.down 
concentration at the downstream interface. One value [M/L3]
only when ( 
dif.flux 
diffusive flux across at the interface of each grid cell.
A vector of length N+1 [M/L2/T]
only when ( 
adv.flux 
advective flux across at the interface of each grid cell.
A vector of length N+1 [M/L2/T]
only when ( 
flux 
total flux across at the interface of each grid cell. A vector
of length N+1 [M/L2/T].
only when ( 
flux.up 
flux across the upstream boundary, positive = INTO model domain. One value [M/L2/T] 
flux.down 
flux across the downstream boundary, positive = OUT of model domain. One value [M/L2/T] 
Note
The advective equation is not checked for mass conservation. Sometimes, this is
not an issue, for instance when v
represents a sinking velocity of
particles or a swimming velocity of organisms.
In others cases however, mass conservation needs to be accounted for.
To ensure mass conservation, the advective velocity must obey certain
continuity constraints: in essence the product of the volume fraction (VF),
interface surface area (A) and advective velocity (v) should be constant.
In sediments, one can use setup.compaction.1D
to ensure that
the advective velocities for the pore water and solid phase meet these
constraints.
In terms of the units of concentrations and fluxes we follow the convention
in the geosciences.
The concentration C
, C.up
, C.down
as well at the rate of
change of the concentration dC
are always expressed per unit of
phase volume (i.e. per unit volume of solid or liquid).
Total concentrations (e.g. per unit volume of bulk sediment) can be obtained by multiplication with the appropriate volume fraction. In contrast, fluxes are always expressed per unit of total interface area (so here the volume fraction is accounted for).
Author(s)
Filip Meysman <filip.meysman@nioz.nl>, Karline Soetaert <karline.soetaert@nioz.nl>
References
Soetaert and Herman (2009). A practical guide to ecological modelling  using R as a simulation platform. Springer
See Also
tran.volume.1D
for a discretisation the transport equation using finite volumes.
tran.2D
, tran.3D
advection.1D
, for more sophisticated advection schemes
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239  ## =============================================================================
## EXAMPLE 1: O2 and OC consumption in sediments
## =============================================================================
# this example uses only the volume fractions
# in the reactive transport term
#====================#
# Model formulation #
#====================#
# Monod consumption of oxygen (O2)
O2.model < function (t = 0, O2, pars = NULL) {
tran < tran.1D(C = O2, C.up = C.ow.O2, D = D.grid,
v = v.grid, VF = por.grid, dx = grid)$dC
reac <  R.O2*(O2/(Ks+O2))
return(list(dCdt = tran + reac))
}
# First order consumption of organic carbon (OC)
OC.model < function (t = 0, OC, pars = NULL) {
tran < tran.1D(C = OC, flux.up = F.OC, D = Db.grid,
v = v.grid, VF = svf.grid, dx = grid)$dC
reac <  k*OC
return(list(dCdt = tran + reac))
}
#======================#
# Parameter definition #
#======================#
# Parameter values
F.OC < 25 # input flux organic carbon [micromol cm2 yr1]
C.ow.O2 < 0.25 # concentration O2 in overlying water [micromol cm3]
por < 0.8 # porosity
D < 400 # diffusion coefficient O2 [cm2 yr1]
Db < 10 # mixing coefficient sediment [cm2 yr1]
v < 1 # advective velocity [cm yr1]
k < 1 # decay constant organic carbon [yr1]
R.O2 < 10 # O2 consumption rate [micromol cm3 yr1]
Ks < 0.005 # O2 consumption saturation constant
# Grid definition
L < 10 # depth of sediment domain [cm]
N < 100 # number of grid layers
grid < setup.grid.1D(x.up = 0, L = L, N = N)
# Volume fractions
por.grid < setup.prop.1D(value = por, grid = grid)
svf.grid < setup.prop.1D(value = (1por), grid = grid)
D.grid < setup.prop.1D(value = D, grid = grid)
Db.grid < setup.prop.1D(value = Db, grid = grid)
v.grid < setup.prop.1D(value = v, grid = grid)
#====================#
# Model solution #
#====================#
# Initial conditions + simulation O2
yini < rep(0, length.out = N)
O2 < steady.1D(y = yini, func = O2.model, nspec = 1)
# Initial conditions + simulation OC
yini < rep(0, length.out = N)
OC < steady.1D(y = yini, func = OC.model, nspec = 1)
# Plotting output, using S3 plot method of package rootSolve"
plot(O2, grid = grid$x.mid, xyswap = TRUE, main = "O2 concentration",
ylab = "depth [cm]", xlab = "", mfrow = c(1,2), type = "p", pch = 16)
plot(OC, grid = grid$x.mid, xyswap = TRUE, main = "C concentration",
ylab = "depth [cm]", xlab = "", mfrow = NULL)
## =============================================================================
## EXAMPLE 2: O2 in a cylindrical and spherical organism
## =============================================================================
# This example uses only the surface areas
# in the reactive transport term
#====================#
# Model formulation #
#====================#
# the numerical model  rate of change = transportconsumption
Cylinder.Model < function(time, O2, pars)
return (list(
tran.1D(C = O2, C.down = BW, D = Da, A = A.cyl, dx = dx)$dC  Q
))
Sphere.Model < function(time, O2, pars)
return (list(
tran.1D(C = O2, C.down = BW, D = Da, A = A.sphere, dx = dx)$dC  Q
))
#======================#
# Parameter definition #
#======================#
# parameter values
BW < 2 # mmol/m3, oxygen conc in surrounding water
Da < 0.5 # cm2/d effective diffusion coeff in organism
R < 0.0025 # cm radius of organism
Q < 250000 # nM/cm3/d oxygen consumption rate/ volume / day
L < 0.05 # cm length of organism (if a cylinder)
# the numerical model
N < 40 # layers in the body
dx < R/N # thickness of each layer
x.mid < seq(dx/2, by = dx, length.out = N) # distance of center to midlayer
x.int < seq(0, by = dx, length.out = N+1) # distance to layer interface
# Cylindrical surfaces
A.cyl < 2*pi*x.int*L # surface at midlayer depth
# Spherical surfaces
A.sphere < 4*pi*x.int^2 # surface of sphere, at each midlayer
#====================#
# Model solution #
#====================#
# the analytical solution of cylindrical and spherical model
cylinder < function(Da, Q, BW, R, r) BW + Q/(4*Da)*(r^2R^2)
sphere < function(Da, Q, BW, R, r) BW + Q/(6*Da)*(r^2R^2)
# solve the model numerically for a cylinder
O2.cyl < steady.1D (y = runif(N), name = "O2",
func = Cylinder.Model, nspec = 1, atol = 1e10)
# solve the model numerically for a sphere
O2.sphere < steady.1D (y = runif(N), name = "O2",
func = Sphere.Model, nspec = 1, atol = 1e10)
#====================#
# Plotting output #
#====================#
# Analytical solution  "observations"
Ana.cyl < cbind(x.mid, O2 = cylinder(Da, Q, BW, R, x.mid))
Ana.spher < cbind(x.mid, O2 = sphere(Da, Q, BW, R, x.mid))
plot(O2.cyl, O2.sphere, grid = x.mid, lwd = 2, lty = 1, col = 1:2,
xlab = "distance from centre, cm",
ylab = "mmol/m3", main = "tran.1D",
sub = "diffusionreaction in a cylinder and sphere",
obs = list(Ana.cyl, Ana.spher), obspar = list(pch = 16, col =1:2))
legend ("topleft", lty = c(1, NA), pch = c(NA, 18),
c("numerical approximation", "analytical solution"))
legend ("bottomright", pch = 16, lty = 1, col = 1:2,
c("cylinder", "sphere"))
## =============================================================================
## EXAMPLE 3: O2 consumption in a spherical aggregate
## =============================================================================
# this example uses both the surface areas and the volume fractions
# in the reactive transport term
#====================#
# Model formulation #
#====================#
Aggregate.Model < function(time, O2, pars) {
tran < tran.1D(C = O2, C.down = C.ow.O2,
D = D.grid, A = A.grid,
VF = por.grid, dx = grid )$dC
reac <  R.O2*(O2/(Ks+O2))*(O2>0)
return(list(dCdt = tran + reac, consumption = reac))
}
#======================#
# Parameter definition #
#======================#
# Parameters
C.ow.O2 < 0.25 # concentration O2 water [micromol cm3]
por < 0.8 # porosity
D < 400 # diffusion coefficient O2 [cm2 yr1]
v < 0 # advective velocity [cm yr1]
R.O2 < 1000000 # O2 consumption rate [micromol cm3 yr1]
Ks < 0.005 # O2 saturation constant [micromol cm3]
# Grid definition
R < 0.025 # radius of the agggregate [cm]
N < 100 # number of grid layers
grid < setup.grid.1D(x.up = 0, L = R, N = N)
# Volume fractions
por.grid < setup.prop.1D(value = por, grid = grid)
D.grid < setup.prop.1D(value = D, grid = grid)
# Surfaces
A.mid < 4*pi*grid$x.mid^2 # surface of sphere at middle of grid cells
A.int < 4*pi*grid$x.int^2 # surface of sphere at interface
A.grid < list(int = A.int, mid = A.mid)
#====================#
# Model solution #
#====================#
# Numerical solution: staedy state
O2.agg < steady.1D (runif(N), func = Aggregate.Model, nspec = 1,
atol = 1e10, names = "O2")
#====================#
# Plotting output #
#====================#
par(mfrow = c(1,1))
plot(grid$x.mid, O2.agg$y, xlab = "distance from centre, cm",
ylab = "mmol/m3",
main = "Diffusionreaction of O2 in a spherical aggregate")
legend ("bottomright", pch = c(1, 18), lty = 1, col = "black",
c("O2 concentration"))
# Similar, using S3 plot method of package rootSolve"
plot(O2.agg, grid = grid$x.mid, which = c("O2", "consumption"),
xlab = "distance from centre, cm", ylab = c("mmol/m3","mmol/m3/d"))
